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LINEARIZATION STABILITY RESULTS AND ACTIVEMEASUREMENTS FOR THE EINSTEIN-SCALAR

FIELD EQUATIONS

YAROSLAV KURYLEV, MATTI LASSAS, GUNTHER UHLMANN

Abstract: We consider linearization stability results for the cou-pled Einstein equations and the scalar field equations for the metric gand scalar fields φ = (φℓ)Lℓ=1 on a 4-dimensional globally hyperbolicLorentzian manifold (M, g). More precisely, we study the Einsteinequations coupled with the scalar field equations and study the sys-tem Ein(g) = T , T = T (g, φ) + F1, and �gφ

ℓ − m2φℓ = F2, wherethe sources F = (F1,F2) correspond to perturbations of the physi-cal fields which we control. The sources F need to be such that thefields (g, φ,F) are solutions of this system and satisfy the conservationlaw divg(T ) = 0. If (gε, φε) solves the above equations, the derivatives

g = ∂εgε|ε=0, φ = φε|ε=0, and f = (f 1, f 2) = ∂εFε|ε=0 solve the lin-earized Einstein equations and the linearized conservation law

1

2gpk∇pf

1kj +

L∑

ℓ=1

f 2ℓ ∂jφℓ = 0, j = 1, 2, 3, 4,

where g = gε|ε=0 and φ = φε|ε=0. In this case we say that (g, φ)and f have the linearization stability property. In linearization sta-bility one ask the converse: If g, φ, and f solve the linearized Einsteinequations and the linearized conservation law, do there exist a familyFε = (F1

ε ,F2ε ) of sources and functions (gε, φε) depending smoothly on

ε ∈ [0, ε0), ε0 > 0, such that (gε, φε) solves the Einstein-scalar fieldequations and the conservation law. Under the condition that the back-

ground fields g and φ vary enough and L ≥ 5, we prove a microlocalversion of this: When Y ⊂M is a 2-dimensional space-like surface and(y, η) is an element of the conormal bundle N∗Y of Y , one can finda linearized source f that is a conormal distibution with respect to the

surface Y with a prescribed principal symbol at (y, η) such that (g, φ)and f have the linearization stability property. This result is provenby constructing a model with adaptive source functions. In this modelthe source term F1 corresponds to e.g. fluid fields consisting of par-ticles which 4-velocity vectors are controlled and F2 contains a termcorresponding to a secondary source function that adapts the changesof g, φ, and F1 so that the physical conservation law is satisfied. Theobtained results can be applied to show that one can send gravitational

Date: May 14, 2014.1

http://arxiv.org/abs/1405.3384v1

2 YAROSLAV KURYLEV, MATTI LASSAS, GUNTHER UHLMANN

waves that propagates along the geodesic determined by (y, η) and thepolarization of this wave can be controlled.AMS classification:83C05, 35J25, 53C50

Contents

1. Introduction and main results 21.1. Formulation of the linearization stability problem 52. Basic properties of the Einstein equations 92.1. Reduced Einstein equation and wave map 92.2. Linearization of Einstein equation and conservation law 133. Analysis for the Einstein-scalar field equations 153.1. Linearized conservation law and harmonicity conditions 164. A model with adaptive source function 204.1. Initial value problem with adaptive source functions 205. Proof of the microlocal linearization stabililty 266. Application: Gravitational wave packets 31Appendix: Motivation of adaptive source functions using

Lagrangian formulation 33References 36

Keywords: Inverse problems, Lorentzian manifolds, Einstein equa-tions, scalar fields, non-linear hyperbolic equations.

1. Introduction and main results

We consider the linearization stability result that is essential for theinverse problems for the non-linear Einstein equations coupled withmatter field equations. In this paper, we consider for the matter fieldsthe simplest possible model, the scalar field equations and study theperturbations of a globally hyperbolic Lorentzian manifold (M, g) ofdimension (1 + 3), where the metric signature of g is (−,+,+,+).

Our problem is related to the following inverse problem: Can anobserver in a space-time determine the structure of the surroundingspace-time by doing measurements near its world line. To study this,we need to produce a large number of measurements, or equivalently,a large number of sources.

LINEARIZATION STABILITY RESULTS 3

Ug I+(p−)

µz,η

FIGURE 1. This is a schematic figure in R1+1. The black vertical

line is the freely falling observer µ([−1, 1]). The rounded black squareis π(Uz0,η0) that is is a neighborhood of z0, and the red curve passingthrough z ∈ π(Uz0,η0) is the time-like geodesic µz,η([−1, 1]). The bound-ary of the domain Ug where we observe waves is shown on blue. Theblack future cone is the set I+g (p

−).

1.0.1. Notations. Let (M, g) be a C∞-smooth (1+3)-dimensional time-orientable Lorentzian manifold. For x, y ∈ M we say that x is in thechronological past of y and denote x ≪ y if x 6= y and there is atime-like path from x to y. If x 6= y and there is a causal path fromx to y, we say that x is in the causal past of y and denote x < y. Ifx < y or x = y we denote x ≤ y. The chronological future I+(p) ofp ∈ M consist of all points x ∈ M such that p ≪ x, and the causalfuture J+(p) of p consist of all points x ∈ M such that p ≤ x. Onedefines similarly the chronological past I−(p) of p and the causal pastJ−(p) of p. For a set A we denote J±(A) = ∪p∈AJ

±(p). We also denoteJ(p, q) := J+(p) ∩ J−(q) and I(p, q) := I+(p) ∩ I−(q). If we need toemphasize the metric g which is used to define the causality, we denoteJ±(p) by J±

g (p) etc.Let γx,ξ(t) = γgx,ξ(t) = expx(tξ) denote a geodesics in (M, g). The

projection from the tangent bundle TM to the base point of a vector isdenoted by π : TM → M . Let LxM denote the light-like directions ofTxM , and L+

xM and L−xM denote the future and past pointing light-

like vectors, respectively. We also denote +g (x) = expx(L

+xM)∪{x} the

union of the image of the future light-cone in the exponential map of(M, g) and the point x.

By [10], an open time-orientable Lorenzian manifold (M, g) is glob-ally hyperbolic if and only if there are no closed causal paths in M andfor all q−, q+ ∈M such that q− < q+ the set J(q−, q+) ⊂ M is compact.We assume throughout the paper that (M, g) is globally hyperbolic.

When g is a Lorentzian metric, having eigenvalues λj(x) and eigen-vectors vj(x) in some local coordinates, we will use also the corre-sponding Riemannian metric, denoted g+ which has the eigenvalues


|λj(x)| and the eigenvectors vj(x) in the same local coordinates. LetBg+(x, r) = {y ∈ M ; dg+(x, y) < r}. Finally, when X is a set, letP (X) = 2X = {Z; Z ⊂ X} denote the power set of X.

1.0.2. Perturbations of a global hyperbolic metric. Let (M, g) be a C∞-smooth globally hyperbolic Lorentzian manifold. We will call g thebackground metric on M and consider its small perturbations. ALorentzian metric g1 dominates the metric g2, if all vectors ξ that arelight-like or time-like with respect to the metric g2 are time-like withrespect to the metric g1, and in this case we denote g2 < g1. As (M, g)is globally hyperbolic, it follows from [34] that there is a Lorentzianmetric g such that (M, g) is globally hyperbolic and g < g. One canassume that the metric g is smooth. We use the positive definite Rie-mannian metric g+ to define norms in the spaces Ck

b (M) of functionswith bounded k derivatives and the Sobolev spaces Hs(M).

By [11], the globally hyperbolic manifold (M, g) has an isometry Φ

to the smooth product manifold (R×N, h), where N is a 3-dimensional

manifold and the metric h can be written as h = −β(t, y)dt2 + κ(t, y)where β : R × N → (0,∞) is a smooth function and κ(t, · ) is a Rie-mannian metric on N depending smoothly on t ∈ R, and the subman-ifolds {t′} × N are C∞-smooth Cauchy surfaces for all t′ ∈ R. Wedefine the smooth time function t : M → R by setting t(x) = t ifΦ(x) ∈ {t} × N . Let us next identify these isometric manifolds, thatis, we denote M = R×N .

For t ∈ R, let M(t) = (−∞, t)×N and, for a fixed t0 > 0 and t1 > t0,let Mj =M(tj), j = 1, 2. Let r0 > 0 be sufficiently small and V(r0) bethe set of metrics g on M1 = (−∞, t1)×N , which C8

b (M1)-distance tog is less that r0 and coincide with g in M(0) = (−∞, 0)×N .

1.0.3. Observation domain U . For g ∈ V(r0), let µg : [−1, 1] → M1 bea freely falling observer, that is, a time-like geodesic on (M, g). Let−1 < s−2 < s−1 < 1 be such that p− = µg(s−1) ∈ {0} ×N . Below, wedenote s− = s−1 and µ = µg.

When z0 = µ(s−2) ∈ M(0) and η0 = ∂sµ(s−2), let Uz0,η0(h) be theopen h-neighborhood of (z0, η0) in the Sasaki metric of (TM, g+). We

use below a small parameter h > 0. For (z, η) ∈ Uz0,η0(2h) we defineon (M, g) a freely falling observer µg,z,η : [−1, 1] → M , such that

µg,z,η(s−2) = z, and ∂sµg,z,η(s−2) = η. We assume that h is so small

that π(Uz0,η0(2h)) ⊂M(0) and for all g ∈ V(r0) and (z, η) ∈ Uz0,η0(2h)the geodesic µg,z,η([−1, 1]) ⊂ M is well defined and time-like. We

denote, see Fig. 1, Uz0,η0 = Uz0,η0(h) and

Ug=⋃

(z,η)∈Uz0,η0

µg,z,η([−1, 1]), U+g = Ug ∩ I

+(µg,z,η(s+)),(1)

and we also denote U = Ug and U+ = U+g .


1.1. Formulation of the linearization stability problem.

1.1.1. Einstein equations. Below, we use the Einstein summation con-vention. The roman indexes i, j, k etc. run usually over the values0, 1, 2, 3 as the greek letters are reserved for other indexes in the sums.The Einstein tensor of a Lorentzian metric g = gjk(x) is

Einjk(g) = Ricjk(g)−1

2(gpq Ricpq(g))gjk.

Here, Ricpq(g) is the Ricci curvature of the metric g. We define thedivergence of a 2-covariant tensor Tjk to be (divgT )k = ∇n(g

njTjk).Let us consider the Einstein equations in presence of matter,

Einjk(g) = Tjk,(2)

divgT = 0,(3)

for a Lorentzian metric g and a stress-energy tensor T related to thedistribution of mass and energy. We recall that by Bianchi’s identitydivg(Ein(g)) = 0 and thus the equation (3), called the conservation lawfor the stress-energy tensor, follows automatically from (2).

1.1.2. Reduced Einstein tensor. Let m ≥ 5, t1 > t0 > 0 and g′ ∈ V(r0)be a Cm-smooth metric that satisfy the Einstein equations Ein(g′) = T ′

on M(t1). When r0 above is small enough, there is a diffeomorphismf :M(t1) → f(M(t1)) ⊂M that is a (g′, g)-wave map f : (M(t1), g

′) →(M, g) and satisfies M(t0) ⊂ f(M(t1)). Here, f : (M(t1), g

′) → (M, g)is a wave map, see [12, Sec. VI.7.2 and App. III, Thm. 4.2], if

�g′,gf = 0 in M(t1),(4)

f = Id, in (−∞, 0)×N,(5)

where �g′,gf = g′ · ∇2f is the wave map operator, and ∇ is the co-variant derivative for the maps (M1, g

′) → (M, g), see [12, formula(VI.7.32)]. The existence and the properties of the map f is dis-cussed below in Subsection 2.1.3. The wave map has the propertythat Ein(f∗g

′) = Eing(f∗g′), where Eing(g) is the g-reduced Einstein

tensor,

(Eingg)pq = −1

2gjk∇j∇kgpq +

1

4(gnmgjk∇j∇kgnm)gpq + Ppq(g, ∇g),

where ∇j is the covariant differentation with respect to the metric g and

Ppq is a polynomial function of gnm, gnm, and ∇jgnm with coefficientsdepending on the metric gnm and its derivatives. Considering the wavemap f as a transformation of coordinates, we see that g = f∗g

′ andT = f∗T

′ satisfy the g-reduced Einstein equation

Eing(g) = T on M(t0).(6)


In the literature, the above is often stated by saying that the reducedEinstein equations (6) is the Einstein equations written with the wave-gauge corresponding to the metric g. The equation (6) is a quasi-linearhyperbolic system of equations for gjk. We emphasize that a solution ofthe reduced Einstein equations can be a solution of the original Einsteinequations only if the stress energy tensor satisfies the conservation law∇g

jTjk = 0. It is usual also to assume that the energy density is

non-negative. For instance, the weak energy condition requires thatTjkX

jXk ≥ 0 for all time-like vectors X. Next, we couple the Einsteinequations with matter fields and formulate an initial value problem forthe g-reduced Einstein equations with abstract sources.

1.1.3. The direct problem. We consider the coupled system of the Ein-stein equation and L scalar field equations with some sources F1 andF2. In these equations we use the wave gauge, that is, the equationsare written for the metric tensor and the scalar fied that are pushedforward with the wave map so that the Einstein tensor of g is equal tothe reduced Einstein tensor Eing(g).

Let g and φ = (φℓ)Lℓ=1 be C∞-background fields on M . Consider

Eing(g) = T, Tjk = Tjk(g, φ) + F1jk, in M0,(7)

Tjk(g, φ) =

L∑

ℓ=1

(∂jφℓ ∂kφℓ −1

2gjkg

pq∂pφℓ ∂qφℓ − V (φℓ)gjk),

�gφℓ − V ′(φℓ) = F2ℓ , ℓ = 1, 2, 3, . . . , L,

g = g and φℓ = φℓ in (−∞, 0)×N

where F1 and F2 are supported in U+g ∩J+

g (p−), V (s) = 1

2m2s2. Above,

�gφ = (−det(g))−1

2∂p((−det(g))1

2 gpq∂qφ). We assume that the back-

ground fields g and φ satisfy the equations (7) with F1 = 0 and F2 = 0.Note that above J+

g (p−) ∩M0 ⊂ J+

g (p−) when g ∈ V(r0).

To obtain a physically meaningful model, we need to assume thatthe physical conservation law in relativity,

∇p(gpkTkj) = 0, for j = 1, 2, 3, 4, where Tkj = Tkj(g, φ) + F1

kj(8)

is satisfied. Here ∇ = ∇g is the connection corresponding to g. InSubsection 2.1.4 we show that for the solution (g, φ) of the system (7)and the conservation law (8) the reduced Einstein tensor Eing(g) willthen be equal to the Einstein tensor Ein(g) .

We mainly need local existence results1 for the system (7). Theglobal existence problem for the related systems has recently attractedmuch interest in the mathematical community and many importantresults been obtained, see e.g. [18, 22, 63, 65, 69, 70].

1In this paper we do not use optimal smoothness for the solutions in classical Ck

spaces or Sobolev space W k,p but just suitable smoothness for which the non-linearwave equations can be easily analyzed using L2-based Sobolev spaces.


We encounter above the difficulty that the source F = (F1,F2) in(7) has to satisfy the condition (8) that depends on the solution g of(7). This makes the formulation of active measurements in relativ-ity difficult. Later, we consider a model where the source term F1

corresponds to e.g. fluid fields consisting of particles whose4-velocityvectors are controlled and F2 contains a term corresponding to a sec-ondary source function that adapts the changes of g, φ, and F1 so thatthe physical conservation law is satisfied.

1.1.4. Linearized equations. We need also to consider the linearizedversion of the equations (7) that have the form (in local coordinates)

�g gjk + Ajk(g, φ, ∂g, ∂φ) = f 1, in M0,(9)

�gφℓ +Bℓ(g, φ, ∂g, ∂φ) = f 2, ℓ = 1, 2, 3, . . . , L,

where Ajk and Bℓ are first order linear differential operators whose

coefficients depend on g and φ. For more explicit formulation, seeSubsection 2.2.1. When gε and φε are solutions of (7) with source Fε

depending smoothly on ε ∈ R such that (gε, φε,Fε)|ε=0 = (g, φ, 0), then

(g, φ, f) = (∂εgε, ∂εφε, ∂εFε)|ε=0 solve (9).Let us consider the concept of the linearization stability (LS) for the

source problems, cf. [33]:

Definition 1.1. Let s0 > 4 and consider a Cs0+4-smooth source f =(f 1, f 2) that is supported in Ug and satisfies the linearized conservationlaw

1

2gpk∇pf

1kj +

L∑

ℓ=1

f 2ℓ ∂jφℓ = 0, j = 1, 2, 3, 4.(10)

Let (g, φ) be the solution of the linearized Einstein equations (9) withsource f . We say that f has the LS-property in Cs0(M0) if there areε0 > 0 and a family Fε = (F1

ε ,F2ε ) of sources, supported in Ugε for all

ε ∈ [0, ε0), and functions (gε, φε) that depend smoothly on ε ∈ [0, ε0) inCs0(M0) such that

(gε, φε) solves the equations (7) and the conservation law (8),(11)

(gε, φε,Fε)|ε=0 = (g, φ, 0), and (g, φ, f) = (∂εgε, ∂εφε, ∂εFε)|ε=0.

In this case, we say that f = (f 1, f 2) has the LS-property with thefamily Fε, ε ∈ [0, ε0).

Note that above (10) is obtained by linearization of the conservationlaw (8).

Next, we consider sources that are conormal distributions. WhenY ⊂ Ug is a 2-dimensional space-like submanifold, consider local co-ordinates defined in V ⊂ M0 such that Y ∩ V ⊂ {x ∈ R

4; xjb1j =

0, xjb2j = 0}, where b1j , b2j ∈ R. Next we slightly abuse the notation

by identifying x ∈ V with its coordinates X(x) ∈ R4. We denote


f ∈ In(Y ), n ∈ R, if in the above local coordinates, f can be writtenas

f(x1, x2, x3, x4) = Re

∫

R2

ei(θ1b1m+θ2b2m)xm

σf (x, θ1, θ2) dθ1dθ2,(12)

where σf (x, θ) ∈ Sn0,1(V ;R

2), θ = (θ1, θ2) is a classical symbol. Afunction c(x, θ) that is n-positive homogeneous in θ, i.e., c(x, sθ) =snc(x, θ) for s > 0, is the principal symbol of f if there is φ ∈ C∞

0 (R2)being 1 near zero such that σf (x, θ) − (1 − φ(θ))c(x, θ) ∈ Sn1

0,1(V ;R2),n1 < n. When η = θ1b1 + θ2b2 ∈ N∗

xY , we say that c(x, η) = c(x, θ) isthe value of the principal symbol of f at (x, η) ∈ N∗Y .

We need a condition that we call microlocal linearization stability:

Definition of the µ-LS (Microlocal linearization stability) prop-erty: We say that the set U+

g , and (M, g) have the microlocal lineariza-tion stability property if the following holds:

Let Y ⊂ U+g be a 2-dimensional space-like submanifold, V ⊂ U+

g anopen local coordinate neighborhood of y ∈ Y with coordinates X : V →R

4, Xj(x) = xj such that X(Y ∩ V ) ⊂ {x ∈ R4; xjb1j = 0, xjb2j = 0}.

Let, in addition, (y, η) ∈ N∗Y be a light-like covector. W ⊂ N∗Y bea conic neighborhood of (y, η), (cjk)

4j,k=1 be a symmetric matrix that

satisfies

glk(y)ηlckj = 0, for all j = 1, 2, 3, 4,(13)

and (dℓ)Lℓ=1 ∈ R

L. Then there is n1 ∈ Z+ such that for any n ∈ Z−,n ≤ −n1 there are f 1

jk ∈ In(Y ), (j, k) ∈ {1, 2, 3, 4}2, and f 2ℓ ∈ In(Y ),

ℓ = 1, 2, . . . , L, supported in V with symbols that are in S−∞ outsidethe neighborhood W of (y, η), and whose principal symbols at (y, η) are

equal to f 1jk(y, η) = cjk and f 2

ℓ (y, η) = dℓ, respectively. Moreover, the

source f = (f 1, f 2) satisfies the linearized conservation law (10) andf has the LS property (11) in Cs1(M0), s1 ≥ 13, with a family Fε,ε ∈ [0, ε0) such that Fε are supported in V .

1.1.5. Main results. We consider the following condition that is validwhen the background fields vary sufficiently:

Condition A: Assume that at any x ∈ U g there is a permutationσ : {1, 2, . . . , L} → {1, 2, . . . , L}, denoted σx, such that the 5 × 5

matrix [Bσjk(φ(x),∇φ(x))]j,k≤5 is invertible, where

[Bσjk(φ(x),∇φ(x))]k,j≤5 =

[( ∂jφℓ(x))ℓ≤5, j≤4

(φℓ(x))ℓ≤5

].

Our main result is that when the Condition A is valid, also thecondition µ-LS is valid:


Theorem 1.2. (Microlocal linearization stability) Let (M, g) be a smooth,globally hyperbolic Lorentzian manifold with a global time function t :M → R and M0 = t−1(−∞, t0). Moreover, let U+

g , Ug ⊂ M0 be the

sets of the form (1) and let (g, φ) satisfy the equation (7) with vanish-ing sources F1 = 0 and F2 = 0. Assume that Condition A is validin Ug. Then the set U+

g and (M, g) have the microlocal linearizationstability property.

2. Basic properties of the Einstein equations

2.1. Reduced Einstein equation and wave map. In this sectionwe review well known results for the Einstein equation and the wavemaps that we will need later.

2.1.1. Geometric considerations. Let us recall some definitions givenin the Introduction. Let (M, g) be a C∞-smooth globally hyperbolicLorentzian manifold and g be a C∞-smooth globally hyperbolic metricon M such that g < g. Let us start by explaining how one can con-struct a C∞-smooth metric g such that g < g and (M, g) is globallyhyperbolic: Let v(x) be an eigenvector corresponding to the negativeeigenvalue of g(x). We can choose a smooth, strictly positive functionη :M → R+ such that

g′ := g − ηv ⊗ v < g.

Then (M, g′) is globally hyperbolic, g′ is smooth and g < g′. Thus wecan replace g by the smooth metric g′ having the same properties thatare required for g.

Recall that there is an isometry Φ : (M, g) → (R × N, h), where

N is a 3-dimensional manifold and the metric h can be written ash = −β(t, y)dt2 + κ(t, y) where β : R × N → (0,∞) is a smoothfunction and κ(t, · ) is a Riemannian metric on N depending smoothlyon t ∈ R. As in the main text we identify these isometric manifolds anddenote M = R×N . Also, for t ∈ R, recall that M(t) = (−∞, t)×N .We use parameters t1 > t0 > 0 and denote Mj = M(tj), j ∈ {0, 1}.We use the time-like geodesic µ = µg, µg : [−1, 1] → M0 on (M0, g)and the set Kj := J+

g (µ(−1)) ∩Mj with µ(−1) ∈ (−∞, t0)×N . Then

J+g (µ(−1)) ∩ Mj is compact. Also, there exists ε0 > 0 such that if

g is a Lorentz metric in M1 such that ‖g − g‖C0b(M1;g+) < ε0, then

g|K1< g|K1

. In particular, this implies that we have J+g (p) ∩M1 ⊂ K1

for all p ∈ K1. Later, we use this property to deduce that when gsatisfies the g-reduced Einstein equations in M1, with a source that issupported in K1 and has small enough norm is a suitable space, theng coincides with g in M1 \ K1 and satisfies g < g.

Let us use local coordinates on M1 and denote by ∇k = ∇Xkthe

covariant derivative with respect to the metric g in the direction Xk =


∂∂xk and by ∇k = ∇Xk

the covariant derivative with respect to themetric g to the direction Xk.

2.1.2. Reduced Ricci and Einstein tensors. Following [32] we recall that

Ricµν(g) = Ric(h)µν (g) +1

2(gµq

∂Γq

∂xν+ gνq

∂Γq

∂xµ)(14)

where Γq = gmnΓqmn,

Ric(h)µν (g) = −1

2gpq

∂2gµν∂xp∂xq

+ Pµν ,(15)

Pµν = gabgpsΓpµbΓ

sνa +

1

2(∂gµν∂xa

Γa + gνlΓlabg

aqgbd∂gqd∂xµ

+ gµlΓlabg

aqgbd∂gqd∂xν

).

Note that Pµν is a polynomial of gjk and gjk and first derivatives of gjk.The harmonic Einstein tensor is

Ein(h)jk (g) = Ric

(h)jk (g)−

1

2gpqRic(h)pq (g) gjk.(16)

The harmonic Einstein tensor is extensively used to study the Ein-stein equations in local coordinates where one can use the Minkowskispace R

4 as the background space. To do global constructions with abackground space (M, g) one uses the reduced Einstein tensor. The g-reduced Einstein tensor Eing(g) and the g-reduced Ricci tensor Ricg(g)are given by

(Eing(g))pq = (Ricg(g))pq −1

2(gjk(Ricgg)jk)gpq,(17)

(Ricg(g))pq = Ricpq(g)−1

2(gpn∇qF

n + gqn∇pFn)(18)

where F n are the harmonicity functions given by

F n = Γn − Γn, where Γn = gjkΓnjk, Γn = gjkΓn

jk,(19)

with Γnjk and Γn

jk being the Christoffel symbols for g and g, correspond-

ingly. Note that Γn depends also on gjk. As Γnjk − Γn

jk is the difference

of two connection coefficients, it is a tensor. Thus F n is tensor (ac-tually, a vector field), implying that both (Ricg(g))jk and (Eing(g))jkare 2-covariant tensors. A direct calculation shows that the g-reducedEinstein tensor is the sum of the harmonic Einstein tensor and a termthat is a zeroth order in g,

(Eing(g))µν = Ein(h)µν (g) +

1

2(gµq

∂Γq

∂xν+ gνq

∂Γq

∂xµ).(20)

We also use the wave operator

(21)

�gφ =4∑

p,q=1

(−det(g(x)))−1

2

∂

∂xp

((−det(g(x)))

1

2 gpq(x)∂

∂xqφ(x)

),


which can be written as

�gφ = gjk∂j∂kφ− gpqΓnpq∂nφ = gjk∂j∂kφ− Γn∂nφ.(22)

2.1.3. Wave maps and reduced Einstein equations. Let us consider themanifold M1 = (−∞, t1) × N with a Cm-smooth metric g′, m ≥ 5,which is a perturbation of the metric g and satisfies the Einstein equa-tion

Ein(g′) = T ′ on M1,(23)

or equivalently,

Ric(g′) = ρ′, ρ′jk = T ′jk −

1

2((g′)nmT ′

nm)g′jk on M1.

Assume also that g′ = g in the domain A, where A = M1 \ K1 and‖g′ − g‖C2

b(M1,g+) < ε0, so that (M1, g

′) is globally hyperbolic. Note

that then T ′ = T in the set A. Then the metric g′ coincides with g inparticular in the set M− = R− ×N

We recall next the considerations of [12]. Let us consider the Cauchyproblem for the wave map f : (M1, g

′) → (M, g), namely

�g′,gf = 0 in M1,(24)

f = Id, in R− ×N,(25)

where M1 = (−∞, t1)×N ⊂M . In (24), �g′,gf = g′ · ∇2f is the wave

map operator, where ∇ is the covariant derivative of a map (M1, g′) →

(M, g), see [12, Ch. VI, formula (7.32)]. In local coordinates X :V → R

4 of V ⊂ M1, denoted X(z) = (xj(z))4j=1 and Y : W → R4

of W ⊂ M , denoted Y (z) = (yA(z))4A=1, the wave map f : M1 → Mhas the representation Y (f(X−1(x))) = (fA(x))4A=1 and the wave mapoperator in equation (24) is given by

(�g′,gf)A(x) = (g′)jk(x)

( ∂

∂xj∂

∂xkfA(x)− Γ′n

jk(x)∂

∂xnfA(x)(26)

+ΓABC(f(x))

∂

∂xjfB(x)

∂

∂xkfC(x)

)

where ΓABC denotes the Christoffel symbols of metric g and Γ′j

kl are theChristoffel symbols of metric g′. When (24) is satisfied, we say that fis wave map with respect to the pair (g′, g).

It follows from [12, App. III, Thm. 4.2 and sec. 4.2.2], that if g′ ∈Cm(M0), m ≥ 5 is sufficiently close to g in Cm(M0), then (4)-(5) has aunique solution f ∈ C0([0, t1];H

m−1(N)) ∩ C1([0, t1];Hm−2(N)). This

comes from the fact that the Christoffel symbols of g′ are in Cm−1(M0).Moreover, when m is even, using [58, Thm. 7] for f and ∂pt f , we seethat the solution f ∈ ∩m−1

p=0 Cp([0, t1];H

m−1−p(N)) ⊂ Cm−3(M0) and fdepends in Cm−3(M0) continuously on g′ ∈ Cm(M0). We note thatthese smoothness results for f are not optimal.


The wave map operator �g′,g is a coordinate invariant operator. Theimportant property of the wave maps is that, if f is wave map withrespect to the pair (g′, g) and g = f∗g

′ then, as follows from (26), theidentity map Id : x 7→ x is a wave map with respect to the pair (g, g)and, the wave map equation for the identity map is equivalent to (cf.[12, p. 162])

Γn = Γn, where Γn = gjkΓnjk, Γn = gjkΓn

jk(27)

where the Christoffel symbols Γnjk of the metric g are smooth functions.

Since g = g′ outside a compact set K1 ⊂ (0, t1) × N , we see thatthis Cauchy problem is equivalent to the same equation restricted tothe set (−∞, t1) × B0, where B0 ⊂ N is an open relatively compactset such that K1 ⊂ (0, t1]×B0 with the boundary condition f = Id on(0, t1] × ∂B0. Moreover, by the uniqueness of the wave map, we havef |M1\K1

= id so that f(K1) ∩M0 ⊂ K0.As the inverse function of the wave map f depends continuously, in

Cm−3b ([0, t1] × N, g+), on the metric g′ ∈ Cm(M0) we can also assume

that ε1 is so small that M0 ⊂ f(M1).Denote next g := f∗g

′, T := f∗T′, and ρ := f∗ρ

′ and define ρ =

T − 12(Tr T )g. Then g is Cm−6-smooth and the equation (23) implies

Ein(g) = T on M0.(28)

Since f is a wave map and g = f∗g′, we have that the identity map is

a (g, g)-wave map and thus g satisfies (27) and thus by the definitionof the reduced Einstein tensor, (18)-(17), we have

Einpq(g) = (Eing(g))pq on M0.

This and (28) yield the g-reduced Einstein equation

(Eing(g))pq = Tpq on M0.(29)

This equation is useful for our considerations as it is a quasilinear,hyperbolic equation on M0. Recall that g coincides with g in M0 \ K0.The unique solvability of this Cauchy problem is studied in e.g. [12,Thm. 4.6 and 4.13], [45].

2.1.4. Relation of the reduced Einstein equations and the original Ein-stein equation. The metric g which solves the g-reduced Einstein equa-tion Eing(g) = T is a solution of the original Einstein equations Ein(g) =

T if the harmonicity functions F n vanish identically. Next we recallthe result that the harmonicity functions vanish on M0 when

(Eing(g))jk = Tjk, on M0,(30)

∇pTpq = 0, on M0,

g = g, on M0 \ K0.


To see this, let us denote Einjk(g) = Sjk, Sjk = gjngkmSnm, and T jk =

gjngkmTnm. Following standard arguments, see [12], we see from (17)that in local coordinates

Sjk − (Eing(g))jk =1

2(gjn∇kF

n + gkn∇jFn − gjk∇nF

n).

Using equations (30), the Bianchi identity ∇pSpq = 0, and the basic

property of Lorentzian connection, ∇kgnm = 0, we obtain

0 = 2∇p(Spq − T pq)

= ∇p(gqk∇kF

p + gpm∇mFq − gpq∇nF

n)

= gpm∇p∇mFq + (gqp∇n∇pF

n − gqp∇p∇nFn)

= gpm∇p∇mFq +W q(F )

where F = (F q)4q=1 and the operator

W : (F q)4q=1 7→ (gqk(∇p∇kFp −∇k∇pF

p))4q=1

is a linear first order differential operator whose coefficients are poly-nomial functions of gjk, g

jk, gjk, gjk and their first derivatives.

Thus the harmonicity functions F q satisfy on M0 the hyperbolicinitial value problem

gpm∇p∇mFq +W q(F ) = 0, on M0,

F q = 0, on M0 \ K0,

and as this initial Cauchy problem is uniquely solvable by [12, Thm.

4.6 and 4.13] or [45], we see that F q = 0 on M0. Thus equations (30)yield that the Einstein equations Ein(g) = T hold on M0.

We note that in the (g, g)-wave map coordinates, where F q = 0, thewave operator (22) has the form

�gφ = gjk∂j∂kφ− gpqΓnpq∂nφ.(31)

Thus, the scalar field equation �gφ−m2φ = 0 does not involve deriva-tives of g.

2.2. Linearization of Einstein equation and conservation law.

2.2.1. Linearized Einstein-scalar field equations. Next we consider thelinearized equations that are obtained as the derivatives of the solutionsof the non-linear, generalized Einstein-matter field equations (7).

Observe that if a family Fε = (F1ε ,F

2ε ) of sources and a family (gε, φε)

of functions are solutions of the non-linear reduced Einstein-scalar fieldequations (7) that depend smoothly on ε ∈ [0, ε0) in C15(M0) and sat-

isfy Fε|ε=0 = 0, (gε, φε)|ε=0 = (g, φ), then ∂εFε|ε=0 = f = (f 1, f 2), and


∂ε(gε, φε)|ε=0 = (g, φ), satisfies the linearized version of the equation(7) that has the form (in the local coordiantes),

�g gjk + Ajk(g, φ, ∂g, ∂φ) = f 1jk, in M0,(32)

�gφℓ +Bℓ(g, φ, ∂g, ∂φ) = f 2ℓ , ℓ = 1, 2, 3, . . . , L.

Here Ajk and Bℓ are first order linear differential operators which co-

efficients depend on g and φ. Let us write these equations in moreexplicit form. We see that the linearized reduced Einstein tensor is inlocal coordinates of the form

epq(g) := ∂ε(Einggε)pq|ε=0

= −1

2gjk∇j∇kgpq +

1

4(gnmgjk∇j∇kgnm)gpq

+Aabnpq ∇ngab +Bab

pq gab,

where Aabnpq (x) and Bab

pq(x) depend on gjk and its derivatives (theseterms can be computed explicitly using (15)). The linearized scalarfield stress-energy tensor is the linear first order differential operator

t(1)pq (g) + t(2)pq (φ) := ∂ε(Tjk(gε, φε))|ε=0

=

L∑

ℓ=1

(∂jφℓ ∂kφℓ + ∂jφℓ ∂kφℓ

−1

2gjkg

pq∂pφℓ ∂qφℓ −1

2gjkg

pq∂pφℓ ∂qφℓ

−1

2gjkg


2gjkg

pq∂pφℓ ∂qφℓ

−m2φℓφℓgjk −1

2m2φ2

ℓ gjk

).

Thus when Fε = (F1ε ,F

2ε ) is a family of sources and (gε, φε) a family

of functions that satisfy the non-linear reduced Einstein-scalar fieldequations (7), the ε-derivatives u = (g, φ) and ∂εFε|ε=0 = f = (f 1, f 2)satisfy in local (g, g)-wave coordinates

epq(g)− t(1)pq (g)− t(2)pq (φ) = f 1pq,(33)

�gφℓ − gnmgkj(∂n∂jφℓ + Γpnj∂pφℓ)gmk −m2φℓ = f 2

ℓ .

We call this linearized Einstein-scalar field equations.

2.2.2. Linearization of the conservation law. Assume that (g, φ) andF = (F1,F2) satisfy equation (7). Then the conservation law (8) gives


for all j = 1, 2, 3, 4 equations (see [12, Sect. 6.4.1])

0 =1

2∇g

p(gpkTjk)

=1

2∇g

p(gpk(Tjk(g, φ) + F1

jk))

=L∑

ℓ=1

(gpk∇gp∂kφℓ) ∂jφℓ − (m2

ℓφℓ∂pφℓ)δpj +

1

2∇g

p(gpkF1

jk)

=L∑

ℓ=1

(gpk∇gp∂kφℓ −m2

ℓφℓ) ∂jφℓ +1

2∇g

p(gpkF1

jk).

This yields by (7)

1

2gpk∇g

pF1jk +

L∑

ℓ=1

F2ℓ ∂jφℓ = 0, j = 1, 2, 3, 4.(34)

Next assume that (gε, φε) and Fε satisfy equation (7) and C1-smoothfunctions of ε ∈ (−ε0, ε0) taking values in H1(N)-tensor fields, and

(gε, φε)|ε=0 = (g, φ) and Fε|ε=0 = 0. Denote (f 1, f 2) = ∂εFε|ε=0. Thenby taking ε-derivative of (34) at ε = 0 we get

1

2gpk∇pf

1jk +

L∑

ℓ=1

f 2ℓ ∂jφℓ = 0, j = 1, 2, 3, 4.(35)

We call this the linearized conservation law.

3. Analysis for the Einstein-scalar field equations

Let we consider the solutions (g, φ) of the equations (7) with source

F . To consider their local existence, let us denote u := (g, φ)− (g, φ).It follows from by [5, Cor. A.5.4] that Kj = J+

g (p−)∩M j is compact.

Since g < g, we see that if r0 above is small enough, for all g ∈V(r0), see subsection 1.0.2, we have g|K1

< g|K1. In particular, we

have J+g (p

−) ∩M1 ⊂ J+g (p

−).

Let us assume that F is small enough in the norm C4b (M0) and that

it is supported in a compact set K = Jg(p−)∩ ([0, t0]×N) ⊂ M0. Then

we can write the equations (7) for u in the form

Pg(u)(u) = F , x ∈M0,(36)

u = 0 in (−∞, 0)×N , where

Pg(u)(u) := gjk(x; u)∂j∂ku(x) +H(x, u(x), ∂u(x)),

(gjk(x; u))4j,k=1 = (gjk(x))−1, where (g, φ) = u+ (g, φ), and (x, v, w) 7→

H(x, v, w) is a smooth function which is a second order polynomialin w with coefficients being smooth functions of v and derivatives ofg, [97]. Note that when the norm of F in C4

b (M0) is small enough,


we have supp (u) ⊂ K. We note that one could also consider non-compactly supported sources or initial data, see [19]. Also, the scalarfield-Einstein system can be considered with much less regularity thatis done below, see [14, 15].

Let s0 ≥ 4 be an even integer. Below we will consider the solutions

u = (g − g, φ − φ) and the sources F as sections of the bundle BL

on M0. We will consider these functions as elements of the section-valued Sobolev spaces Hs(M0;B

L) etc. Below, we omit the bundle BL

in these notations and denote Hs(M0;BL) = Hs(M0). We use the same

convention for the spaces

Es =

s⋂

j=0

Cj([0, t0];Hs−j(N)), s ∈ N.

Note that Es ⊂ Cp([0, t0] × N) when 0 ≤ p < s − 2. Local existenceresults for (36) follow from the standard techniques for quasi-linearequations developed e.g. in [45] or [58], or [90, Section 9]. These yieldthat when F is supported in the compact set K and ‖F‖Es0 < c0,where c0 > 0 is small enough, there exists a unique function u satisfyingequation (36) on M0 with the source F . Moreover,

‖u‖Es0 ≤ C1‖F‖Es0 .(37)

For a detailed analysis, see Appendix B in [61].

3.1. Linearized conservation law and harmonicity conditions.

3.1.1. Lagrangian distributions. Let us recall definition of the conormaland Lagrangian distributions that we will use below. Let X be a man-ifold of dimension n and Λ ⊂ T ∗X \ {0} be a Lagrangian submanifold.Let φ(x, θ), θ ∈ R

N be a non-degenerate phase function that locallyparametrizes Λ. We say that a distribution u ∈ D′(X) is a Lagrangiandistribution associated with Λ and denote u ∈ Im(X ; Λ), if in localcoordinates u can be represented as an oscillatory integral,

u(x) =

∫

RN

eiφ(x,θ)a(x, θ) dθ,(38)

where a(x, θ) ∈ Sm+n/4−N/2(X ;RN), see [38, 49, 78].In particular, when S ⊂ X is a submanifold, its conormal bundle

N∗S = {(x, ξ) ∈ T ∗X \ {0}; x ∈ S, ξ ⊥ TxS} is a Lagrangian sub-manifold. If u is a Lagrangian distribution associated with Λ1 whereΛ1 = N∗S, we say that u is a conormal distribution.

Let us next consider the case when X = Rn and let (x1, x2, . . . , xn) =

(x′, x′′, x′′′) be the Euclidean coordinates with x′ = (x1, . . . , xd1), x′′ =

(xd1+1, . . . , xd1+d2), x′′′ = (xd1+d2+1, . . . , xn). If S1 = {x′ = 0} ⊂ R

n,Λ1 = N∗S1 then u ∈ Im(X ; Λ1) can be represented by (38) with N = d1and φ(x, θ) = x′· θ.


Next we recall the definition of Ip,l(X ; Λ1,Λ2), the space of the dis-tributions u in D′(X) associated to two cleanly intersecting Lagrangianmanifolds Λ1,Λ2 ⊂ T ∗X \{0}, see [21, 38, 78]. These classes have beenwidely used in the study of inverse problems, see [16, 29]. Let us startwith the case when X = R

n.Let S1, S2 ⊂ R

n be the linear subspaces of codimensions d1 and d1+d2, respectively, S2 ⊂ S1, given by S1 = {x′ = 0}, S2 = {x′ = x′′ = 0}.Let us denote Λ1 = N∗S1, Λ2 = N∗S2. Then u ∈ Ip,l(Rn;N∗S1, N

∗S2)if and only if

u(x) =

∫

Rd1+d2

ei(x′·θ′+x′′·θ′′)a(x, θ′, θ′′) dθ′dθ′′,

where the symbol a(x, θ′, θ′′) belongs in the product type symbol classSµ1,µ2(Rn; (Rd1 \ 0)× R

d2) that is the space of function a ∈ C∞(Rn ×R

d1 × Rd2) that satisfy

|∂γx∂αθ′∂

βθ′′a(x, θ

′, θ′′)| ≤ CαβγK(1 + |θ′|+ |θ′′|)µ1−|α|(1 + |θ′′|)µ2−|β|(39)

for all x ∈ K, multi-indexes α, β, γ, and compact sets K ⊂ Rn. Above,

µ1 = p+ l − d1/2 + n/4 and µ2 = −l − d2/2.When X is a manifold of dimension n and Λ1,Λ2 ⊂ T ∗X \ {0}

are two cleanly intersecting Lagrangian manifolds, we define the classIp,l(X ; Λ1,Λ2) ⊂ D′(X) to consist of locally finite sums of distributionsof the form u = Au0, where u0 ∈ Ip,l(Rn;N∗S1, N

∗S2) and S1, S2 ⊂ Rn

are the linear subspace of codimensions d1 and d1+d2, respectively, suchthat S2 ⊂ S1, and A is a Fourier integral operator of order zero with acanonical relation Σ for which Σ ◦ (N∗S1)

′ ⊂ Λ′1 and Σ ◦ (N∗S2)

′ ⊂ Λ′2.

Here, for Λ ⊂ T ∗X we denote Λ′ = {(x,−ξ) ∈ T ∗X ; (x, ξ) ∈ Λ}, andfor Σ ⊂ T ∗X × T ∗X we denote Σ′ = {(x, ξ, y,−η); (x, ξ, y, η) ∈ Σ}.

In most cases, below X = M . We denote then Ip(M ; Λ1) = Ip(Λ1)and Ip,l(M ; Λ1,Λ2) = Ip,l(Λ1,Λ2). Also, I(Λ1) = ∪p∈RI

p(Λ1).By [38, 78], microlocally away from Λ1 and Λ0,

Ip,l(Λ0,Λ1) ⊂ Ip+l(Λ0 \ Λ1) and Ip,l(Λ0,Λ1) ⊂ Ip(Λ1 \ Λ0),(40)

respectively. Thus the principal symbol of u ∈ Ip,l(Λ0,Λ1) is well de-fined on Λ0 \Λ1 and Λ1 \Λ0. We denote I(Λ0,Λ1) = ∪p,q∈RI

p,q(Λ0,Λ1).Below, when Λj = N∗Sj, j = 1, 2 are conormal bundles of smooth

cleanly intersecting submanifolds Sj ⊂ M of codimension mj , wheredim (M) = n, we use the traditional notations,

Iµ(S1) = Iµ+m1/2−n/4(N∗S1), Iµ1,µ2(S1, S2) = Ip,l(N∗S1, N∗S2),(41)

where p = µ1 + µ2 +m1/2 − n/4 and l = −µ2 −m2/2, and call suchdistributions conormal distributions associated to S1 or product typeconormal distributions associated to S1 and S2, respectively. By [38],Iµ(X ;S1) ⊂ Lp

loc(X) for µ < −m1(p− 1)/p, 1 ≤ p <∞.For the wave operator �g on the globally hyperbolic manifold (M, g)

Char (�g) is the set of light-like vectors with respect to g, and (y, η) ∈


Θx,ξ if and only if there is t ∈ R such that (y, η) = (γgx,b(t), γgx,b(t))

where γgx,b is a light-like geodesic with respect to the metric g with the

initial data (x, b) ∈ TM , and a = η♭, b = ξ♭.Let P = �g + B0 + Bj∂j , where B0 and Bj are tensors. Then

P is a classical pseudodifferential operator of real principal type andorder m = 2 on M , and [78], see also [64], P has a parametrix Q ∈Ip,l(∆′

T ∗M ,ΛP ), p = 12−m, l = −1

2, where ∆T ∗M = N∗({(x, x); x ∈

M}) and Λg ⊂ T ∗M × T ∗M is the Lagrangian manifold associated tothe canonical relation of the operator P , that is,

Λg = {(x, ξ, y,−η); (x, ξ) ∈ Char (P ), (y, η) ∈ Θx,ξ},(42)

where Θx,ξ ⊂ T ∗M is the bicharacteristic of P containing (x, ξ). When(M, g) is a globally hyperbolic manifold, the operator P has a causalinverse operator, see e.g. [5, Thm. 3.2.11]. We denote it by P−1 and by[78], we have P−1 ∈ I−3/2,−1/2(∆′

T ∗M ,Λg). We will repeatedly use thefact (see [38, Prop. 2.1]) that if F ∈ Ip(Λ0) and Λ0 intersects Char(P )transversally so that all bicharacterestics of P intersect Λ0 only finitelymany times, then (�g + B0 + Bj∂j)

−1F ∈ Ip−3/2,−1/2(Λ0,Λ1) whereΛ′

1 = Λg ◦ Λ′0 is called the flowout from Λ0 on Char(P ), that is,

Λ1 = {(x,−ξ); (x, ξ, y,−η) ∈ Λg, (y, η) ∈ Λ0}.

3.1.2. The linearized Einstein equations and the linearized conserva-tion law. We will below consider sources F = εf(x) and solution uεsatisfying (36), where f = (f (1), f (2)).

We consider the linearized Einstein equations and the linearized waveu(1) = ∂εuε|ε=0. It satisfies the linearized Einstein equations (9) thatwe write as

�gu(1) + V (x, ∂x)u

(1) = f ,(43)

where v 7→ V (x, ∂x)v is a linear first order partial differential operatorwith coefficients depending on the derivatives of g.

Assume that Y ⊂M0 is a 2-dimensional space-like submanifold andconsider local coordinates defined in V ⊂ M0. Moreover, assume thatin these local coordinates Y ∩V ⊂ {x ∈ R

4; xjbj = 0, xjb′j = 0}, where

b′j ∈ R and let f = (f (1), f (2)) ∈ In(Y ), n ≤ n0 = −17, be defined by

f(x1, x2, x3, x4) = Re

∫

R2

ei(θ1bm+θ2b′m)xm

σf (x, θ1, θ2) dθ1dθ2.(44)

Here, we assume that σf (x, θ), θ = (θ1, θ2) is a BL-valued classicalsymbol and we denote the principal symbol of f by c(x, θ) = σp(f)(x, θ),

or component-wise, ((c(1)jk (x, θ))

4j,k=1, (c

(2)ℓ (x, θ))Lℓ=1). When x ∈ Y and

ξ = (θ1bm + θ2b′m)dx

m so that (x, ξ) ∈ N∗Y , we denote the value ofthe principal symbol f at (x, ξ) by c(x, ξ) = c(x, θ), that is component-

wise, c(1)jk (x, ξ) = c

(1)jk (x, θ) and c

(2)ℓ (x, ξ) = c

(2)ℓ (x, θ). We say that this

is the principal symbol of f at (x, ξ), associated to the phase function


φ(x, θ1, θ2) = (θ1bm + θ2b′m)x

m. The above defined principal symbolscan be defined invariantly, see [42].

We will below consider what happens when f = (f (1), f (2)) ∈ In(Y )satisfies the linearized conservation law (10). Roughly speaking, thesefour linear conditions imply that the principal symbol of the source f

satisfies four linear conditions. Furthermore, the linearized conserva-tion implies that also the linearized wave u(1) produced by f satisfiesfour linear conditions that we call the linearized harmonicity condi-tions, and finally, the principal symbol of the wave u(1) has to satisfyfour linear conditions. Next we explain these conditions in detail.

When (10) is valid, we have

glkξlc(1)kj (x, ξ) = 0, for j ≤ 4 and ξ ∈ N∗

xY .(45)

We say that this is the linearized conservation law for the principalsymbols. Note that Iµ(Y ) ⊂ Cs(M0) when s ≤ −µ − 3. We will lateruse such indexes µ so that we can use s = 13.

3.1.3. A harmonicity condition for the linearized solutions. Assumethat (g, φ) satisfy equations (7) and the conservation law (8) is valid.The conservation law (8) and the g-reduced Einstein equations (7) im-ply, see e.g. [12, 90], that the harmonicity functions Γj = gnmΓj

nm

satisfy

gnmΓjnm = gnmΓj

nm.(46)

Next we denote u(1) = (g1, φ1) = (g, φ), and discuss the implicationsof this for the metric component g of the solution of the linearizedEinstein equations.

We do next calculations in local coordinates of M0 and denote ∂k =∂

∂xk . Direct calculations show that hjk = gjk√−det(g) satisfies ∂kh

kq =

−Γqknh

nk. Then (46) implies that

∂khkq = −Γq

knhnk.(47)

We call (47) the harmonicity condition for the metric g.Assume now that gε and φε satisfy (7) with source F = εf where

ε > 0 is a small parameter. We define hjkε = gjkε√

−det(gε) and denote

gjk = ∂ε(gε)jk|ε=0, gjk = ∂ε(gε)

jk|ε=0, and hjk = ∂εhjkε |ε=0.

The equation (47) yields then2

∂khkq = −Γq

knhnk.(48)

A direct computation shows that

hab = (−det(g))1/2κab,

2The treatment on this de Donder-type gauge condition is known in the folkloreof the field. For a similar gauge condition to (48) in harmonic coordinates, see [72,pages 6 and 250], or [67, formulas 107.5, 108.7, 108.8], or [46, p. 229-230].


where κab = gab − 12gabgqpg

pq. Thus (48) gives

∂a((−det(g))1/2κab) = −Γbac(−det(g))1/2κac(49)

that implies ∂aκab + κnbΓa

an + κanΓban = 0, or equivalently,

∇aκab = 0.(50)

We call (50) the linearized harmonicity condition for g. Writing thisfor g, we obtain

−gan∂agnj +1

2gpq∂j gpq = mpq

j gpq(51)

where mj depend on gpq and its derivatives. On similar conditionsfor the polarization tensor, see [85, form. (9.58) and example 9.5.a, p.416].

3.1.4. Properties of the principal symbols of the waves. Let K ⊂ M0

be a light-like submanifold of dimension 3 that in local coordinatesX : V → R

4, xk = Xk(y) is given by K ∩ V ⊂ {x ∈ R4; bkx

k = 0},

where bk ∈ R are constants. Assume that the solution u(1) = (g, φ) ofthe linear wave equation (43) with the right hand side vanishing in Vis such that u(1) ∈ Iµ(K) with µ ∈ R. Below we use µ = n− 3

2where

n ∈ Z−, n ≤ n0 = −18. Let us write gjk as an oscillatory integral usinga phase function ϕ(x, θ) = bkx

kθ, and a symbol ajk(x, θ) ∈ Snclas(R

4,R),

gpq(x1, x2, x3, x4) = Re

∫

R

ei(θbmxm)apq(x, θ) dθ,(52)

where n = µ + 12. We denote the (positively homogeneous) principal

symbol of apq(x, θ) by σp(gpq)(x, θ). When x ∈ K and ξ = θbkdxk so

that (x, ξ) ∈ N∗K, we denote the value of σp(gpq) at (x, θ) by ajk(x, ξ),that is, ajk(x, ξ) = σp(gpq)(x, θ).

Then, if gjk satisfies the linearized harmonicity condition (46), itsprincipal symbol ajk(x, ξ) satisfies

−gmn(x)ξmvnj +1

2ξj(g

pq(x)vpq) = 0, vpq = apq(x, ξ),(53)

where j = 1, 2, 3, 4 and ξ = θbkdxk ∈ N∗

xK. If (53) holds, we saythat the harmonicity condition for the symbol is satisfied for a(x, ξ) at(x, ξ) ∈ N∗K.

4. A model with adaptive source function

4.1. Initial value problem with adaptive source functions. Letus define some physical fields and introduce a model as a system ofpartial differential equations. Later we will motivate this system bydiscussion of the corresponding Lagrangians, but we postpone this dis-cussion to a last section as it is not completely rigorous.

We assume that there are C∞-background fields g, φ, on M .


We consider a Lorentzian metric g on M0 and φ = (φℓ)Lℓ=1 where φℓ

are scalar fields on M0 = (−∞, t0)×N .Let P = Pjk(x)dx

jdxk be a symmetric tensor on M0, correspondingbelow to a direct perturbation to the stress energy tensor, and Q =(Qℓ(x))

Kℓ=1 where Qℓ(x) are real-valued functions on M0, where K ≥

L+ 1. We denote by V(φℓ;Sℓ) the potential functions of the fields φℓ,

V(φℓ;Sℓ) =1

2m2(φℓ +

1

m2Sℓ

)2.(54)

These potentials depend on the source variables Sℓ. The way how Sℓ,called below the adaptive source functions, depend on other fields is

explained later. We assume that there are smooth background fields P

and Q. For a while we consider the case when P = 0 and Q = 0, anddiscuss later the generalization to non-vanishing background fields.

Using the φ and P fields, we define the stress-energy tensor

Tjk =L∑

ℓ=1


2gjkg

pq∂pφℓ ∂qφℓ − V(φℓ;Sℓ)gjk) + Pjk.(55)

We assume that P and Q are supported on K = J+g (p

−) ∩M0. Let

us represent the stress energy tensor (55) in the form

Tjk = Pjk + Zgjk +Tjk(g, φ), Z = −

L∑

ℓ=1

(Sℓφℓ +1

2m2S2ℓ ),

Tjk(g, φ) =

L∑

ℓ=1


2gjkg


2m2φ2

ℓgjk),

where we call Z the stress energy density caused by the sources Sℓ.Now we are ready to formulate the direct problem for the adaptive

Einstein-scalar field equations. Let g and φ satisfy

Eing(g) = Pjk + Zgjk +Tjk(g, φ), Z = −L∑

ℓ=1

(Sℓφℓ +1

2m2S2ℓ ),(56)

�gφℓ − V ′(φℓ;Sℓ) = 0 in M0, ℓ = 1, 2, 3, . . . , L,

Sℓ = Sℓ(g, φ,∇φ,Q,∇Q,P,∇gP ), in M0,

g = g, φℓ = φℓ, in (−∞, 0)×N.

Above, V ′(φ; s) = ∂φV(φ; s) so that V ′(φℓ;Sℓ) = m2φℓ+Sℓ. We assume

that the background fields g, φ, satisfy these equations with Q = 0 and

P = 0.We consider here P = (Pjk)

4j,k=1 and Q = (Qℓ)

Kℓ=1 as fields that we

can control and call those the controlled source fields. Local existenceof the solution for small sources P and Q is considered in Appendix B.

To obtain a physically meaningful model, we need to consider howthe adaptive source functions Sℓ should be chosen so that the physical


conservation law in relativity

∇k(gkpTpq) = 0(57)

is satisfied. Here ∇ = ∇g is the connection corresponding to the metricg.

We note that the conservation law is a necessary condition for theequation (56) to have solutions for which Eing(g) = Ein(g), i.e., thatthe solutions of (56) are solutions of the Einstein field equations.

The functions Sℓ(g, φ,∇φ,Q,∇Q,P,∇gP ) model the devices that we

use to perform active measurements. Thus, even though the ConditionS below may appear quite technical, this assumption can be viewed asthe instructions on how to build a device that can be used to measurethe structure of the spacetime far away. Outside the support of themeasurement device (that contain the union of the supports of Q andP ) we have just assumed that the standard coupled Einstein-scalarfield equations hold, c.f. (58). We can consider them in the form

Sℓ(g, φ,∇φ,Q,∇Q,P,∇gP ) = Qℓ + S2nd

ℓ (g, φ,∇φ,Q,∇Q,P,∇gP )

for ℓ = 1, 2, . . . , L where Qℓ are the primary sources and S2ndℓ , that

depend also on Qℓ with ℓ = L + 1, L + 2, . . . , K, corresponds to theresponse of the measurement device that forces the conservation law tobe valid.

The solution (g, φ) of (56) is a solution of the equations (7) when wedenote

F1jk = Pjk + Zgjk,

F2ℓ = V ′(φℓ;Sℓ)− V ′(φℓ; 0) = Sℓ.

Our next goal is to construct suitable adaptive source functions Sℓ andconsider what kind of sources F1 and F2 of the above form can beobtained by varying P and Q.

We will consider adaptive source functions Sℓ satisfying the followingconditions:

Condition S:The adaptive source functions Sℓ(g, φ,∇φ,Q,∇Q,P,∇

gP ) have the fol-lowing properties:

(i) Denoting c = ∇φ, C = ∇gP , and H = ∇Q we assume thatSℓ(g, φ, c, Q,H, P, C) are smooth non-linear functions, of the point-wise values gjk(x), φ(x),∇φ(x), Q(x),∇Q(x), P (x), and ∇gP (x), de-

fined near (g, φ, c, Q,H, P, C) = (g, φ,∇φ, 0, 0, 0, 0), that satisfy

Sℓ(g, φ, c, 0, 0, 0, 0) = 0.(58)

(ii) We assume that Sℓ is independent of P (x) and the dependencyof S on ∇gP and ∇Q is only due to the dependency in the term


gpk∇gp(Pjk + Zgjk) = gpk∇g

pPjk + ∇gjQK , associated to the divergence

of the perturbation of T , that is, there exist functions Sℓ so that

Sℓ(g, φ, c, Q,H, P, C) = Sℓ(g, φ, c, Q,R), R = (gpk∇gp(Pjk +QKgjk))

4j=1.

Below, denote R = gpk∇pPjk + ∇jQK . Note that we still are consid-

ering the case when Q = 0 and P = 0 so that R = 0, too. This impliesthat for the background fields that adaptive source functions Sℓ vanish.

To simplify notations, we also denote below Sℓ just by Sℓ and indicatethe function which we use by the used variables in these functions.

Below we will denote Q = (Q′, QK), Q′ = (Qℓ)

K−1ℓ=1 . There are exam-

ples when the background fields (g, φ) and the adaptive source functionsSℓ exists and satisfy the Condition S.

Our next aim is to prove the following:

Theorem 4.1. Let L ≥ 5 and assume that Q = 0 and P = 0 so

that R = 0. Moreover, assume that Condition A is valid. Then forall permutations σ : {1, 2, . . . , L} → {1, 2, . . . , L} there exists functionsSℓ,σ satisfying Condition S such that

(i) For all x ∈ Ug,σ the differential of

Sσ(g, φ,∇φ, Q,R) = (Sℓ,σ(g, φ,∇φ, Q,R))Lℓ=1

with respect to Q and R, that is, the map

DQ,RSσ(g(x), φ(x),∇φ(x), Q,R)|Q=Q(x),R=R(x) : RK+4 → R

L(59)

is surjective.

(ii) The adaptive source functions Sσ are such that for (Qℓ)Kℓ=1 and

(Pjk) that are sufficiently close to Q = 0 and P = 0 in the C3b (M0)-

topology and supported in Ug,σ the equations (56) with source functionsSσ have a unique solution (g, φ) and the conservation law (57) is valid.

(iii) Under the same assumptions as in (ii), when (g, φ) is a solutionof (56) with the controlled source functions P and Q, we have QK = Z.This means that the physical field Z can be directly controlled.

Proof. As one can enumerate the ℓ-indexes of the fields φℓ as onewishes, it is enough to prove the claim with one σ. We consider belowthe case when σ = Id.

Consider a symmetric (0,2)-tensor P and a scalar functions Qℓ thatare C3-smooth and compactly supported in Ug,σ. Let [Pjk(x)]

4j,k=1 be

the coefficients of P in local coordinates and Q(x) = (Qℓ(x))Lℓ=1.

To obtain adaptive required adaptive source functions, let us startto consider implications of the conservation law (57). To this end,consider C2-smooth functions Sℓ(x) on Ug,σ.


Note that since [∇p,∇n] = [∂p, ∂n] = 0 (see [12, Sect. III.6.4.1]),

∇p(gpjTjk(g, φ) =

L∑

ℓ=1

∇pgpj(∂jφℓ ∂kφℓ −

1

2gjkg

nm∂nφℓ ∂mφℓ −1

2m2φ2

ℓgjk)

=

L∑

ℓ=1

(gpj∇p∂jφℓ) ∂kφℓ +

L∑

ℓ=1

(gpj∂jφℓ∇p∂kφℓ)

−1

2

L∑

ℓ=1

δpk(gnm(∇p∂nφℓ) ∂mφℓ + gnm∂nφℓ (∇p∂mφℓ))−

L∑

ℓ=1

m2δpkφℓ∂pφℓ

=

L∑

ℓ=1

(gpj∇p∂jφℓ) ∂kφℓ +

L∑

ℓ=1

(gpj∂jφℓ (∇p∂kφℓ))

−1

2

L∑

ℓ=1

(gnm∂mφℓ (∇n∂kφℓ) + gnm∂nφℓ (∇m∂kφℓ))−

L∑

ℓ=1

m2φℓ∂kφℓ

=L∑

ℓ=1

(gpj∇p∂jφℓ −m2φℓ) ∂kφℓ.

Thus conservation law (57) gives for all j = 1, 2, 3, 4 equations

0 = ∇gp(g

pkTjk)

= ∇gp(g

pk(Tjk(g, φ) + Pjk + Zgjk))

=L∑

ℓ=1

(gpk∇gp∂kφℓ) ∂jφℓ − (m2

ℓφℓ∂pφℓ)δpj −∇g

p(gpkgjk(Sℓφℓ+

1

2m2S2ℓ ) + gpkPjk)

=L∑

ℓ=1

(gpk∇gp∂kφℓ −m2

ℓφℓ) ∂jφℓ −∇gp(g

pkgjk(Sℓφℓ+1

2m2S2ℓ ) + gpkPjk)

=

L∑

ℓ=1

Sℓ ∂jφℓ −∇gp(g

pkgjk(Sℓφℓ+1

2m2S2ℓ )) + gpk∇g

pPjk

=

(L∑

ℓ=1

Sℓ ∂jφℓ

)− ∂j

(L∑

ℓ=1

Sℓφℓ+1

2m2S2ℓ

)+ gpk∇g

pPjk.

Summarizing, the conservation law gives(

L∑

ℓ=1

Sℓ ∂jφℓ

)− ∂j

(L∑

ℓ=1

Sℓφℓ +1

2m2S2ℓ

)+ gpk∇g

pPjk = 0,(60)

for j = 1, 2, 3, 4.Recall that the field Z has the definition

L∑

ℓ=1

Sℓφℓ +1

2m2S2ℓ = −Z.(61)


Then, the conservation law (57) holds if we have

L∑

ℓ=1

Sℓ ∂jφℓ = −gpk∇gpV jk, V jk = (Pjk + gjkZ),(62)

for j = 1, 2, 3, 4.Equations (61) and (62) give together five point-wise equations for

the functions S1, . . . , SL.Recall that we consider here the case when σ = Id. By Condition

A, at any x ∈ Ug,σ that the 5 × 5 matrix (Bσjk(φ(x),∇φ(x)))j,k≤5 is

invertible, where

(Bσjk(φ(x),∇φ(x)))j,k≤5 =

(( ∂jφℓ(x))j≤4, ℓ≤5

(φℓ(x))ℓ≤5

).

We consider a RK valued function Q(x) = (Q′(x), QK(x)), where

Q′ = (Qℓ)K−1ℓ=1 .

Also, below Rj = gpk∇gpV jk, V jk = Pjk + gjkZ and we require that

the identity

QK = Z(63)

holds.Motivated by equations (61), (62), and (63), our next aim is to con-

sider a point x ∈ Ug,σ, and construct scalar functions Sσ,ℓ(φ,∇φ,Q′, QK , R, g),

ℓ = 1, 2, . . . , L that satisfy

5∑

ℓ=1

Sσ,ℓ(φ,∇φ,Q′, QK , R, g) ∂jφℓ = −Rj −

L∑

ℓ=6

Qσ,ℓ ∂jφℓ,(64)

5∑

ℓ=1

Sσ,ℓ(φ,∇φ,Q′, QK , R, g)φℓ = −

(QK +

L∑

ℓ=6

Qσ,ℓφℓ +

+

L∑

ℓ=1

1

2m2Sσ,ℓ(φ,∇φ,Q

′, QK , R, g)2).

Let

(Yσ(φ,∇φ))(x) = ψ(x)(Bσ(φ,∇φ))−1, for x ∈ Ug,σ,

(Yσ(φ,∇φ))(x) = 0, for x 6∈ Ug,σ,

where ψ ∈ C∞0 (Ug,σ) has value 1 in supp (Q) ∪ supp (P ).

Then we define Sσ,ℓ = Sσ,ℓ(g, φ,∇φ,Q′, QK , R), ℓ = 1, 2, . . . , L, to be

the solution of the system

(Sσ,ℓ)ℓ≤5 = Yσ(φ,∇φ)

((−Rj −

∑Lℓ=6Qσ,ℓ ∂jφℓ)j≤4

−QK −∑L

ℓ=6Qσ,ℓφℓ −∑L

ℓ=11

2m2S2σ,ℓ

)(65)

(Sσ,ℓ)ℓ≥6 = (Qℓ)ℓ≥6.


When Q and R are sufficiently small, this equation can be solved point-wisely, at each point x ∈ Ug,σ, using iteration by the Banach fixed pointtheorem.

Let

(Kσjk(φ(x),∇φ(x)))j,k≤5 =

(( ∂jφℓ(x))j≤4, 6≤ℓ≤L

(φℓ(x))6≤ℓ≤L

).

Then we see that the differential of Sσ = (Sσ,ℓ)Lℓ=1 with respect to

(Q′, QK , R) at (Q,R) = (0, 0), that is,

(66)

DQ′,QK ,RSσ(g, φ,∇φ, Q′, QK , R)|Q=0,R=0 : R

K+4 → RL,

(Q′, QK , R) 7→ −

(Yσ(φ,∇φ) Yσ(φ,∇φ)K(φ,∇φ)

0 Iσ

)(

RQK

)

Q′

,

is surjective, where Iσ = [δk,j+5]k≤K−1, j≤L−5, ∈ R(K−1)×(L−5). Hence

(i) is valid.By their construction, the functions Sσ = (Sσ,ℓ)

Lℓ=1 satisfy the equa-

tions (61) and (62) for all x ∈ Ug,σ and also equation (63) holds.Hence (iii) is valid.Above, the equation (62) is valid by the construction of functions

(Sℓ)Lℓ=1. Thus the conservation law is valid. This proves (ii). �

Note that as the adaptive source functions Sℓ were constructed inTheorem 4.1 using inverse function theorem, the results of Theorem

4.1 are valid also if Q and P are sufficiently small non-vanishing fields

and g and φ satisfy the Einstein scalar field equations (56) with these

background fields. Next we return to the case when P = 0 and Q = 0.

5. Proof of the microlocal linearization stabililty

Below we consider the case when P = 0 and Q = 0 and use theadaptive source functions Sℓ constructed in Theorem 4.1 and its proof.

Assume that Y ⊂M0 is a 2-dimensional space-like submanifold andconsider local coordinates defined in V ⊂ M0. Moreover, assume thatin these local coordinates Y ∩V ⊂ {x ∈ R

4; xjbj = 0, xjb′j = 0}, whereb′j ∈ R and let p ∈ In(Y ), n ≤ n0 = −17, be defined by

pjk(x1, x2, x3, x4) = Re

∫

R2

ei(θ1bm+θ2b′m)xm

vjk(x, θ1, θ2) dθ1dθ2.(67)

Here, we assume that vjk(x, θ), θ = (θ1, θ2) are classical symbols andwe denote their principal symbols by σp(pjk)(x, θ). When x ∈ Y andξ = (θ1bm+ θ2b

′m)dx

m so that (x, ξ) ∈ N∗Y , we denote the value of the

principal symbol σp(p) at (x, θ1, θ2) by v(a)jk (x, ξ), that is, v

(a)jk (x, ξ) =

σp(pjk)(x, θ1, θ2), and say that it is the principal symbol of pjk at (x, ξ),


associated to the phase function φ(x, θ1, θ2) = (θ1bm + θ2b′m)x

m. Theabove defined principal symbols can be defined invariantly, see [42].

We assume that also q′, z ∈ In(Y ) have representations (67) withclassical symbols. Below we consider symbols in local coordinates.Let us denote the principal symbols of p,q′, z ∈ In(Y ) by v(a)(x, ξ),

w(a)1 (x, ξ), w

(a)2 (x, ξ), respectively and let v(b)(x, ξ) and w

(b)2 (x, ξ) denote

the sub-principal symbols of p and z, correspondingly, at (x, ξ) ∈ N∗Y .We will below consider what happens when (pjk + zgjk) ∈ In(Y )

satisfies

glk∇gl (pjk + zgjk) ∈ In(Y ), j = 1, 2, 3, 4.(68)

Note that a priori this function is only in In+1(Y ), so the assumption

(68) means that glk∇gl (pjk + zgjk) is one degree smoother than it a

priori should be.When (68) is valid, we say that the leading order of singularity of

the wave satisfies the linearized conservation law. This corresponds tothe assumption that the principal symbol of the sum of divergence ofthe first two terms appearing in the stress energy tensor on the righthand side of (56) vanishes.

By [42], the identity (68) is equivalent to the vanishing of the prin-cipal symbol on N∗Y , that is,

glkξl(v(a)kj (x, ξ) + gkj(x)w

(a)2 (x, ξ)) = 0, for j ≤ 4 and ξ ∈ N∗

xY .(69)

We say that this is the linearized conservation law for the principalsymbol of R.

Let us consider source fields that have the form Q′ε = ((Qε)ℓ)

K−1ℓ=1 =

εq′, (Qε)K = εz and Pε = εp. We denote q = (q′, z). We assume that

q′, z, and p are supported in V ⊂⊂ U .Let uε = (gε, φε) be the solution of (56) with source Pε and Qε.

Then uε depends C4-smoothly on ε and (gε, φε)|ε=0 = (g, φ). Denote

∂ε(gε, φε)|ε=0 = (g, φ). When ε0 is small enough, Pε and Qε are sup-ported in Ugε for all ε ∈ (0, ε0).

Let

Rε = gpkε ∇gεp ((Pε)jk + gεjk(Qε)K)

and

(Sε)ℓ = Sℓ(gε, φε,∇φε, Q′ε, (Qε)K , Rε),

where Sℓ are the adaptive source functions constructed in Theorem 4.1and its proof.

Then Sε|ε=0 = 0 and ∂εSε|ε=0 = S satisfy

(70)

Sℓ = DQ′,QK ,RSℓ(g, φ,∇φ, Q′, QK , R)

∣∣∣Q′=0,QK=0,R=0

q′

z

r

,


where r = gpk∇p(pjk + gjkz).

The functions u = (g, φ) satisfy the linearized Einstein-scalar fieldequation (33). The linearized Einstein-scalar field equation (33) is

epq(g)− t(1)pq (g)− t(2)pq (φ) = f1pq

�gφℓ − gnmgkj(∂n∂jφℓ + Γp

nj∂pφℓ)gmk −m2φℓ = f2ℓ ,

where

f1pq = ppq − (

L∑

ℓ=1

Sℓφℓ)gpq, where − (

L∑

ℓ=1

Sℓφℓ) = z,(71)

f2ℓ = Sℓ.

By Theorem 4.1 (ii), uε = (gε, φε) satisfy the conservation law (57).

This implies that u = (g, φ) satisfies the linearized Einstein-scalar fieldequation (33) and linearized of the conservation law (10) is valid, too.The linearized of the conservation law (10) gives, by considerationsbefore (71),

(L∑

ℓ=1

f2ℓ ∂jφℓ

)+ gpk∇pf

1kj = 0.(72)

Below, we use the adaptive source functions Sℓ constructed in The-orem 4.1 and its proof.

We see that

f = F (x;p,q) = (F (1)(x;p,q), F (2)(x;p,q))(73)

has by formulas (70) and (71) and Condition S the form

F(1)jk (x;p,q) = pjk + z(x)gjk(x)(74)

and

F (2)(x;p,q) = M(2)q′ + L(2)z+N j

(2)glk∇l(pjk + zgjk),(75)

where

M(2) =M(2)(φ(x), ∇φ(x), g(x)),

L(2) = L(2)(φ(x), ∇φ(x), g(x)),

N j(2) = N j

(2)(φ(x), ∇φ(x), g(x))

are, in local coordinates, matrices whose elements are smooth functions

of φ(x), ∇φ(x), and g(x). By Thm. 4.1 (i), the union of the image

spaces of the matrices M(2)(x) and L(2)(x), and N j(2)(x), j = 1, 2, 3, 4,

span the space RL for all x ∈ Ug,σ.

Consider n ∈ Z, t0, s0 > 0, Y = Y (x0, ζ0; t0, s0), K = K(x0, ζ0; t0, s0)and (x, ξ) ∈ N∗Y (to recall the definitions of these notations, see for-mula (77) and definitions below it). Let Z = Z(x, ξ) be set of the

values of the principal symbol f(x, ξ) = (f1(x, ξ), f2(x, ξ)), at (x, ξ),


of the source f = (f1(x), f2(x)) ∈ In(Y ) that satisfy the linearizedconservation law for principal symbols (13).

We use the following auxiliary result:

Lemma 5.1. Assume the the Condition A is satisfied and Q = 0 and

P = 0. Let k0 ≥ 8, s1 ≥ k0 + 5, and Y ⊂ Ug be a 2-dimensional space-like submanifold and y ∈ Y , ξ ∈ N∗

yY , and let W be a conic neigh-borhood of (y, ξ) in T ∗M . Also, let y ∈ Ug,σ with some permutation σ.Let us consider an open, relatively compact local coordinate neighbor-hood V ⊂ Ug,σ ∩ U+

g of y such that in the coordinates X : V → R4,

Xj(x) = xj, we have X(Y ∩ V ) ⊂ {x ∈ R4; xjb1j = 0, xjb2j = 0}.

Let n1 ∈ Z+ be sufficiently large and n ≤ −n1. Let us considerp,q′, z ∈ In+1(Y ), supported in V , that have classical symbols with

principal symbols v(a)(x, ξ), w(a)1 (x, ξ), w

(a)2 (x, ξ), correspondingly, at

(x, ξ) ∈ N∗Y . Moreover, assume that the principal symbols of p andz satisfy the linearized conservation law for the principal symbols, thatis, (69), at all N∗Y ∩ N∗K and assume that they vanish outside theconic neighborhood W of (y, ξ) in T ∗M . Let f = (f 1, f 2) ∈ In+1(Y ) begiven by (70) and (71).

Then the principal symbol f(y, ξ) = (f1(y, ξ), f2(y, ξ)) of the sourcef at (y, ξ) is the set Z = Z(y, ξ). Moreover, by varying p,q′, z so thatthe linearized conservation law (69) for principal symbols is satisfied,

the principal symbol f(y, ξ) at (y, ξ) achieves all values in the (L + 6)dimensional space Z.

Proof. Let us use local coordinates X : V → R4 where V ⊂ M0 is

a neighborhood of x. In these coordinates, let v(b)(x, ξ) and w(b)2 (x, ξ)

denote the sub-principal symbols of p and z, respectively, at (x, ξ) ∈

N∗Y . Moreover, let v(c)j (x, ξ) = ∂

∂xj v(a)(x, ξ) and d

(c)j (x, ξ) = ∂

∂xj d(a)2 (x, ξ),

j = 1, 2, 3, 4 be the x-derivatives of the principal symbols and let usdenote

v(c)(x, ξ) = (v(c)j (x, ξ))4j=1, d(c)(x, ξ) = (d

(c)j (x, ξ))4j=1.

Let f = (f1, f2) = F (x;p,q) be defined by (74) and (75). Whenthe principal symbols of p,q′, z ∈ In(Y ) are such that the linearizedconservation law (69) for principal symbols is satisfied, we see that

f ∈ In(Y ) has the principal symbol f(x, ξ) = (f1(x, ξ), f2(x, ξ)) at(x, ξ), given by

f1(x, ξ) = s1(x, ξ),

f2(x, ξ) = s2(x, ξ),


where we use the notations

s1(x, ξ) = (v(a) + gw(a)2 )(x, ξ),(76)

s2(x, ξ) =(M(2)(x)w

(a)1 + J(2)(v

(c) + gd(c)) +

+L(2)w(a)2 +N j

(2) glkξl(v

(b)1 + gw

(b)2 )jk

)(x, ξ).

Here, roughly speaking, the J(2) term appears when the ∇-derivativesin R hit to the symbols of the conormal distributions having the form(67). We emphasize that here the symbols s1(x, ξ) and s2(x, ξ) are welldefined objects (in fixed local coordinates) also when the linearizedconservation law (69) for principal symbols is not valid. When (69) isvalid, f ∈ In(Y ) and s1(x, ξ) and s1(x, ξ) coincide with the principalsymbols of f1 and f2.

Observe that the map (c(b)jk ) 7→ (glkξlc

(b)jk )

4j=1, defined as Symm(R4×4) →

R4, is surjective. Denote

m(a) = (v(a) + gw(a)2 )(x, ξ),

m(b) = (v(b) + gw(b)2 )(x, ξ),

m(c) = (v(c) + gd(c))(x, ξ).

As noted above, by (66), the union of the image spaces of the matricesM(2)(x) and L(2)(x), and N j

(2)(x), j = 1, 2, 3, 4, span the space RL for

all x ∈ U . Hence the map

A : (m(a), m(b), m(c), w1, w(a)2 )|(x,ξ) 7→ (s1(x, ξ), s2(x, ξ)),

given by (76), considered as a map A : Y = (Symm(R4×4))1+1+4 ×R

K × R → Symm(R4×4) × RL, is surjective. Let X be the set of ele-

ments (m(a)(x, ξ), m(b)(x, ξ), m(c)(x, ξ), w(a)1 (x, ξ), w

(a)2 (x, ξ)) ∈ Y where

m(a)(x, ξ) = (v(a)+gw(a)2 )(x, ξ) is such that the pair (v(a)(x, ξ), w

(a)2 (x, ξ))

satisfies the linearized conservation law for principal symbols, see (69).Then X has codimension 4 in Y , we see that the image A(X ) has inSymm(R4×4)× R

L co-dimension less or equal to 4.By (72)) and considerations above it, we have that f satisfies the

linearized conservation law (10). This implies that its principal symbol

A((m(a)(x, ξ), m(b)(x, ξ), m(c)(x, ξ), w(a)1 (x, ξ), w

(a)2 (x, ξ))) has to satisfy

the linearized conservation law for principal symbols (13) and henceA(X ) ⊂ Z. As Z has codimension 4, this and the above prove thatA(X ) = Z. �

Now we are ready to prove the microlocal stability result for theEinstein-scalar field equation (7). Note that the claim of the followingtheorem does not involve the adaptive source functions constructed inTheorem 4.1 as these functions are needed only as an auxiliary tool inthe proof.

Next we prove Theorem 1.2.


Proof. Let σ ∈ Σ(K) be such that y ∈ Ug,σ. Let p and q be the func-tions constructed in Lemma 5.1. We can assume that these functionsare supported in W0 = V0 ∩ V ∩ Ug,σ. Let Pε = εp and Qε = εq besources depending on ε ∈ R and uε = (gε, φε) be the solution of (56)with the sources Pε and Qε. Also, let

F1ε = Pε + Zεgε, Zε = −(

L∑

ℓ=1

Sεℓφ

εℓ +

1

2m2(Sε

ℓ )2),

(F2ε )ℓ = Sε

ℓ ,

where

Sεℓ = Sℓ(gε, φε,∇φε, Qε,∇Qε, Pε,∇

gεPε),

where Sℓ are the adaptive source functions constructed in Theorem 4.1and its proof.

By (58) also Sεℓ and the family Fε, ε ∈ [0, ε0] of non-linear sources

are supported in V0 and we have shown that uε = (gε, φε) and Fε satisfythe reduced Einstein-scalar field equation (7) and the conservation law(57). This proves Theorem 1.2. �

6. Application: Gravitational wave packets

Next we consider a distorted plane wave whose singular support isconcentrated near a geodesic. These waves, sketched in Fig. 1(Right),propagate near the geodesic γx0,ζ0([t0,∞)) and are singular on a surfaceK(x0, ζ0; t0, s0), defined below in (77), that is a subset of the lightcone +

g (x′), x′ = γx0,ξ0(t0). The parameter s0 gives a “width” of the

wave packet and when s0 → 0, its singular support tends to the setγx0,ζ0([2t0,∞)). Next we will define these wave packets.

y0 ΣΣ1

y′

FIGURE 2. This is a schematic figure in the space R3. It describes

the location of a distorted plane wave (or a piece of a spherical wave)u at different time moments. This wave propagates near the geodesicγx0,ζ0((0,∞)) ⊂ R

1+3, x0 = (y0, t0) and is singular on a subset of alight cone emanated from x′ = (y′, t′). The piece of the distorted planewave is sent from the surface Σ ⊂ R

3, it starts to propagate, and at alater time its singular support is the surface Σ1.


We define the 3-submanifold K(x0, ζ0; t0, s0) ⊂ M0 associated to(x0, ζ0) ∈ L+(M0, g), x0 ∈ Ug and parameters t0, s0 ∈ R+ as

K(x0, ζ0; t0, s0) = {γx′,η(t) ∈M0; η ∈ W, t ∈ (0,∞)},(77)

where (x′, ζ ′) = (γx0,ζ0(t0), γx0,ζ0(t0)) and W ⊂ L+x′(M0, g) is a neighbor-

hood of ζ ′ consisting of vectors η ∈ L+x′(M0) satisfying ‖η − ζ ′‖g+ < s0.

Note that K(x0, ζ0; t0, s0) ⊂ +g (x

′) is a subset of the light cone start-

ing with x′ = γx0,ξ0(t0) and that it is singular at the point x′. LetS = {x ∈ M0; t(x) = t(γx0,ζ0(2t0))} be a Cauchy surface which in-tersects γx0,ζ0(R) transversally at the point γx0,ζ0(2t0). When t0 > 0is small enough, Y (x0, ζ0; t0, s0) = S ∩ K(x0, ζ0; t0, s0) is a smooth 2-dimensional space-like surface that is a subset of Ug.

Let Λ(x0, ζ0; t0, s0) be the Lagranginan manifold that is the flowoutfrom N∗Y (x0, ζ0; t0, s0) ∩N

∗K(x0, ζ0; t0, s0) on Char(�g) in the futuredirection. When Kreg ⊂ K = K(x0, ζ0; t0, s0) is the set of points xthat have a neighborhood W such that K ∩W a smooth 3-dimensionalsubmanifold, we have N∗Kreg ⊂ Λ(x0, ζ0; t0, s0). Below, we representlocally the elements w ∈ Bx in the fiber of the bundle B as a (10 +L)-dimensional vector, w = (wm)

10+Lm=1 .

Lemma 6.1. Let n1 be a sufficiently large integer, n ≤ −n1, t0, s0 > 0,Y = Y (x0, ζ0; t0, s0), K = K(x0, ζ0; t0, s0), Λ1 = Λ(x0, ζ0; t0, s0), and(x, ξ) ∈ N∗Y ∩ Λ1. Assume that f = (f1, f2) ∈ In(Y ), is a BL-valuedconormal distribution that is supported in a neighborhood V ⊂ M0

of γx0,ζ0 ∩ Y = {γx0,ζ0(2t0)} and has a R10+L-valued classical sym-

bol. Denote the principal symbol of f by f(x, ξ) = (fk(x, ξ))10+Lk=1 , and

assume that the symbol of f vanishes near the light-like directions inN∗Y \N∗K.

Let (g, φ) be a solution of the linear wave equation (43) with thesource f . Then u(1), considered as a vector valued Lagrangian distri-bution on the set M0 \ Y , satisfies u(1) ∈ In−3/2(M0 \ Y ; Λ1), and itsprincipal symbol a(y, η) = (aj(y, η))

10+Lj=1 at (y, η) ∈ Λ1 is given by

aj(y, η) =

10+L∑

k=1

Rkj (y, η, x, ξ)fk(x, ξ),(78)

where the pairs (x, ξ) and (y, η) are on the same bicharacteristics of�g, and x < y, that is, ((x, ξ), (y, η)) ∈ Λ′

g, and in addition, (x, ξ) ∈

N∗Y ∩N∗K. Moreover, the matrix (Rkj (y, η, x, ξ))

10+Lj,k=1 is invertible.

We call the solution u(1) a distorted plane wave that is associated tothe submanifold K(x0, ζ0; t0, s0).

Proof. As noted above (42), the parametrix of the scalar wave equa-tion satisfies (�g +V (x,D))−1 ∈ I−3/2,−1/2(∆′

T ∗M0,Λg), where V (x,D)

is a 1st order differential operator, ∆T ∗M0is the conormal bundle of


the diagonal of M0 ×M0 and Λg is the flow-out of the canonical re-lation of �g. A geometric representation for its kernel is given in[64]. An analogous result holds for the matrix valued wave operator,�gI + V (x,D), when V (x,D) is a 1st order differential operator, thatis, (�gI + V (x,D))−1 ∈ I−3/2,−1/2(∆′

T ∗M0,Λg), see [78] and [24]. By

[38, Prop. 2.1], this yields u(1) ∈ In−3/2(Λ1) and the formula (78) whereR = (Rk

j (y, η, x, ξ))10+Lj,k=1 is obtained by solving a system of ordinary dif-

ferential equation along a bicharacteristic curve. Making similar con-siderations for the adjoint of the (�gI + V (x,D))−1, i.e., consideringthe propagation of singularities using reversed causality, we see thatthe matrix R is invertible. �

Let BLy be the fiber of the bundle BL at y and Sy,η be the space of

the elements in BLy satisfying the harmonicity condition for the symbols

(53) at (y, η). Let (x, ξ) ∈ N∗Y and Cx,ξ the set of elements in BLx that

satisfy the linearized conservation law for symbols, i.e., equation (45).Let n ≤ n0, t0, s0 > 0, Y = Y (x0, ζ0; t0, s0), K = K(x0, ζ0; t0, s0),

Λ1 = Λ(x0, ζ0; t0, s0), and v ∈ Cx,ξ. By Condition µ-SL, there is aconormal distribution f ∈ In(Y ) = In(N∗Y ) such that f satisfies the

linearized conservation law (10) and the principal symbol f(y, η) of f ,

defined on N∗Y , satisfies f(x, sη) = f(x, η)sn for s > 0. Moreover,by Condition µ-SL there is a family of sources Fε, ε ∈ [0, ε0) such

that ∂εFε|ε=0 = f and a solution uε + (g, φ) of the Einstein equationswith the source Fε that depend smoothly on ε and uε|ε=0 = 0. Thenu = ∂εuε|ε=0 ∈ In−3/2(M0 \ Y ; Λ1).

Let (x, ξ) ∈ N∗Y ∩ Λ1 (y, η) ∈ T ∗M0, y 6∈ Y be a light-like co-

vector such that (y, η) ∈ Θx,ξ. Since u = (g, φ) satisfies the lin-earized harmonicity condition (46), the principal symbol a(y, η) =(a1(y, η), a2(y, η)) of u satisfies a(y, η) ∈ Sy,η, This shows that the

map R = R(y, η, x, ξ), given by R : f(x, ξ) 7→ a(y, η) that is defined inLemma 6.1, satisfies R : Cx,ξ → Sy,η. Since R is one-to-one and thelinear spaces Cx,ξ and Sy,η have the same dimension, we see that

R : Cx,ξ → Sy,η(79)

is a bijection. Hence, when f ∈ In(Y ) varies so that the linearizedconservation law (45) for the principal symbols is satisfied, the principalsymbol a(y, η) at (y, η) of the solution u of the linearized Einsteinequation achieves all values in the (L+ 6) dimensional space Sy,η.

Appendix: Motivation of adaptive source functions using

Lagrangian formulation

To motivate the system (56) of partial differential equations, we givein this appendix a non-rigorous discussion.

Following [12, Ch. III, Sect. 6.4, 7.1, 7.2, 7.3] and [3, p. 36] westart by considering the Lagrangians, associated to gravity, scalar fields


φ = (φℓ)Lℓ=1 and non-interacting fluid fields, that is, the number density

four-currents n = (nκ(x))Jκ=1 (where each nκ is a vector field, see [3,

p. 33]). We consider also products of vector fields nκ(x) and 12-density

|det(g)|1/2 that denote pκ,

pjκ(x)

∂

∂xj= nj

κ(x)|det(g)|1/2∂

∂xj.(80)

see [25, p. 53]. Also, ρ = (−gjknjκn

kκ)

1/2 corresponds to the energydensity of the fluid. Below, we use the variation of density with respectto the metric,

δ

δgjk(

J∑

κ=1

(−gnmpnκp

mκ )

1/2) = −J∑

κ=1

1

2(−gnmp

nκp

mκ )

−1/2pjκp

kκ(81)

= −J∑

κ=1

1

2ρnj

κnkκ |det(g)|1/2.

Due to this, we denote

(82)

P =J∑

κ=1

1

2ρnκ

jnκkdx

j ⊗ dxk, where nκk = gkin

iκ = gkip

iκ|det(g)|−1/2.

Below, we consider a model for g, φ, and p. We also add in to themodel a Lagrangian associated with some scalar valued source fieldsS = (Sℓ)

Lℓ=1 and Q = (Qk)

Kk=1. We consider action corresponding to

the coupled Lagrangians

A =

∫

M

(Lgrav(x) + Lfields(x) + Lsource(x)

)dVg(x),

Lgrav =1

2R(g),

Lfields =

L∑

ℓ=1

(−

1

2gjk∂jφℓ ∂kφℓ − V(φℓ;Sℓ)

)+

+J∑

κ=1

(−

1

2(−gjkp

jκp

kκ)

1

2

)|det(g)|−

1

2 ,

Lsource = εHε(g, S,Q,p, φ),

where R(g) is the scalar curvature, dVg = (−det(g))1/2dx is the volumeform on (M, g),

V(φℓ;Sℓ) =1

2m2(φℓ +

1

m2Sℓ

)2(83)

are energy potentials of the scalar fields φℓ that depend on Sℓ, andHε(g, S,Q,p, φ) is a function modeling the measurement device weuse. We assume that Hε is bounded and its derivatives with respect to


S,Q,p are very large (like of order O((ε)−2)) and its derivatives withrespect of g and φ are bounded when ε > 0 is small. We note thatthe above Lagrangian for the fluid fields is the sum of the single fluidLagrangians. where for all fluids the master function Λ(s) = s1/2, thatis, the energy density of each fluid is given by ρ = Λ(−gjkn

jnk).Forfluid Lagrangians, see the discussions in [3, p. 33-37], [12, Ch. III,Sect. 8], [25, p. 53], and [98] and [30, p. 196].

When we compute the critical points of the Lagrangian L and neglectthe O(ε)-terms, the equation δA

δg= 0, together with formulas (81) and

(82), give the Einstein equations with a stress-energy tensor Tjk definedin (55). The equation δA

δφ= 0 gives the wave equations with sources Sℓ.

We assume that O(ε−1) order equations obtained from the equation( δAδS, δAδQ, δAδp) = 0 fix the values of the scalar functions Q and the fields

pκ, κ = 1, 2, . . . , J , and moreover, yield for the sources S = (Sℓ)Lℓ=1

equations of the form

Sℓ = Sℓ(g, φ,∇φ,Q,∇Q,P,∇gP )(84)

where P is given by (82). Let us aslo write (84) using different nota-tions, as

Sℓ = Qℓ + S2ndℓ (g, φ,∇φ,Q,∇Q,P,∇gP ).

Summarizing, we have obtained, up to the above used approximations,the model (56). However, note that above the field P is not directlycontrolled but instead, we control p and the value of the field P isdetermined by the solution n and formula (82). In this sense P is notcontrolled, but an observed field.

Above, the function Hε models the way the measurement deviceworks. Due to this we will assume that Hε and thus functions Sℓ

may be quite complicated. The interpretation of the above is that ineach measurement event we use a device that fixes the values of thescalar functions Q, p, and gives the equations for S2nd that tell howthe sources of the φ-fields adapt to these changes so that the physicalconservation laws are satisfied.

Acknowledgements. The authors express their gratitude to MSRI,the Newton Institute, the Fields Institute and the Mittag-Leffler Insti-tute, where parts of this work have been done.

YK was partly supported by EPSRC and the AXA professorship atthe Mittag-Leffler Institute. ML was partly supported by the FinnishCentre of Excellence in Inverse Problems Research 2012-2017. GU waspartly supported by NSF, a Clay Senior Award at MSRI, a Chancel-lor Professorship at UC Berkeley, a Rothschild Distinguished VisitingFellowship at the Newton Institute, the Fondation de Sciences Mathé-matiques de Paris, and a Simons Fellowship.


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Yaroslav Kurylev, UCL; Matti Lassas, University of Helsinki; Gun-

ther Uhlmann, University of Washington, and University of Helsinki.

[email protected], [email protected], [email protected]

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